# Electromagnetic units of measure

In physics many are unit systems for electrical and magnetic quantities have been developed. The SI has prevailed for the most part; At least in theoretical physics, however, some authors prefer the Gaussian variant of the CGS system.

Not only the specific selection, but also the number of basic quantities in a physical system of units is arbitrary: One can eliminate basic quantities from a unit system by choosing instead the proportionality factor in a linear “law of nature” as a dimensionless number. In theoretical atomic and particle physics, for example, one works with a system of units that has a single base quantity, since vacuum, the speed of light and Planck's quantum of action are set equal to 1.

## Basics

Electromagnetic quantities are linked to mechanical quantities by several linear laws. The following relationships are particularly relevant for the choice of the unit system:

The Coulomb law , which gives the force F between two point charges Q 1 and Q 2 at a distance  r ,

${\ displaystyle F = k_ {1} \ cdot {\ frac {Q_ {1} Q_ {2}} {r ^ {2}}}}$

the Ampère force law , the force F between two currents I 1 and I 2 -carrying conductors of the length L in the distance d indicates

${\ displaystyle F = k_ {2} \ cdot 2 {\ frac {I_ {1} \ cdot I_ {2} \ cdot L} {d}}}$;
${\ displaystyle \ nabla \ times {\ vec {E}} = - k_ {3} \ cdot \ partial {\ vec {B}} / \ partial t.}$

The Coulomb force exerted by static charges and the Lorentz force exerted by moving charges can be compared directly with one another; the context contains the speed of light c . ${\ displaystyle \ textstyle {\ frac {k_ {1}} {k_ {2}}} = c ^ {2}}$

This leaves two independent proportionality constants and , which allow an arbitrary choice of an electrical and a magnetic base unit. In systems of measurement that explicitly reduce the electromagnetic quantities to mechanical quantities, one can choose both constants as dimensionless numbers or as mechanical quantities of arbitrary dimensions. ${\ displaystyle k_ {1}}$${\ displaystyle k_ {3}}$

### Electrostatic system of units

The Electrostatic System of Units (abbreviated esu , or ESU for e lectro s tatic u nits ) is the former way, that is ; so is . ${\ displaystyle k_ {1} = k_ {3} = 1}$${\ displaystyle k_ {2} = c ^ {- 2}}$

### Electromagnetic system of units

The Electromagnetic System of Units (abbreviated emu , or EMU for e lectro m agnetic u nits ) sets ; so is . ${\ displaystyle k_ {2} = k_ {3} = 1}$${\ displaystyle k_ {1} = c ^ {2}}$

### Gaussian system of units

The Gaussian system of units chooses like the electrostatic system and thus ; it then assumes that the speed of light appears in the Maxwell equations in a perfectly symmetrical form. ${\ displaystyle k_ {1} = 1}$${\ displaystyle k_ {2} = c ^ {- 2}}$${\ displaystyle k_ {3} = c ^ {- 1}}$

### Heaviside-Lorentz unit system

The Heaviside-Lorentz system of units also chooses , but differs from the Gaussian system in the choice . The factor 4π anticipates an integration over the solid angle; it makes Coulomb's law more complicated, but simplifies the calculation of the capacitance of a plate capacitor. ${\ displaystyle k_ {3} = c ^ {- 1}}$${\ displaystyle k_ {1} = \ textstyle {\ frac {1} {4 \ pi}}}$

### International system of units

The International System of Units (SI) has an additional base unit, the ampere . This results in a further constant, the magnetic field constant , as well as the electrical field constant linked to it . The SI sets , and . ${\ textstyle c}$ ${\ textstyle \ mu _ {0}}$ ${\ textstyle \ varepsilon _ {0} = 1 / \ mu _ {0} c ^ {2}}$${\ textstyle k_ {1} = {\ frac {1} {4 \ pi \ varepsilon _ {0}}}}$${\ textstyle k_ {2} = {\ frac {\ mu _ {0}} {4 \ pi}}}$${\ textstyle k_ {3} = 1}$

Before the change in the SI system of units in 2019 , the ampere was defined by the Amperes law of force . Therefore the magnetic field constant had an exact value , and since the definition of the meter is specified, it also had an exact value. With the current definition of the ampere, and are measured quantities with measurement uncertainty. ${\ textstyle \ mu _ {0} = 4 \ pi \ cdot 10 ^ {- 7} {\ frac {\ mathrm {N}} {\ mathrm {A} ^ {2}}}}$${\ textstyle c}$${\ textstyle \ varepsilon _ {0}}$${\ textstyle \ mu _ {0}}$${\ textstyle \ varepsilon _ {0}}$

## Important formulas

The following table gives an overview of the form of the most important equations in electrodynamics in the various systems of units:

theme formula Constant K (or , ) in the following system of units: ${\ displaystyle K_ {a}}$${\ displaystyle K_ {b}}$
SI electro
statically
electro
-magnetic
Gauss Heaviside-
Lorentz
Coulomb's
law
${\ displaystyle {\ vec {F}} = K {\ frac {q_ {1} q_ {2}} {r ^ {2}}} {\ vec {e}} _ {r}}$ ${\ displaystyle {\ frac {1} {4 \ pi \ varepsilon _ {0}}}}$ ${\ displaystyle 1}$ ${\ displaystyle c ^ {2}}$ ${\ displaystyle 1}$ ${\ displaystyle {\ frac {1} {4 \ pi}}}$
Force effect of
parallel currents
${\ displaystyle F = 2K {\ frac {I_ {1} I_ {2} L} {d}}}$ ${\ displaystyle {\ frac {\ mu _ {0}} {4 \ pi}}}$ ${\ displaystyle c ^ {- 2}}$ ${\ displaystyle 1}$ ${\ displaystyle c ^ {- 2}}$ ${\ displaystyle {\ frac {1} {4 \ pi c ^ {2}}}}$
Biot-Savart
law
${\ displaystyle {\ vec {B}} = Kq {\ frac {{\ vec {v}} \ times {\ vec {e}} _ {r}} {r ^ {2}}}}$ ${\ displaystyle {\ frac {\ mu _ {0}} {4 \ pi}}}$ ${\ displaystyle c ^ {- 2}}$ ${\ displaystyle 1}$ ${\ displaystyle c ^ {- 1}}$ ${\ displaystyle {\ frac {1} {4 \ pi c}}}$
Lorentz force ${\ displaystyle {\ vec {F}} = Kq {\ vec {v}} \ times {\ vec {B}}}$ ${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle c ^ {- 1}}$ ${\ displaystyle c ^ {- 1}}$
Dielectric polarization ${\ displaystyle {\ vec {D}} = K_ {a} {\ vec {E}} + K_ {b} {\ vec {P}}}$ ${\ displaystyle \ varepsilon _ {0}}$, ${\ displaystyle 1}$ ${\ displaystyle 1}$, ${\ displaystyle 4 \ pi}$ ${\ displaystyle c ^ {- 2}}$, ${\ displaystyle 4 \ pi}$ ${\ displaystyle 1}$, ${\ displaystyle 4 \ pi}$ ${\ displaystyle 1}$, ${\ displaystyle 1}$
magnetization ${\ displaystyle {\ vec {B}} = K_ {a} {\ vec {H}} + K_ {b} {\ vec {M}}}$ ${\ displaystyle \ mu _ {0}}$, ${\ displaystyle \ mu _ {0}}$ ${\ displaystyle c ^ {- 2}}$, ${\ displaystyle 4 \ pi c ^ {- 2}}$ ${\ displaystyle 1}$, ${\ displaystyle 4 \ pi}$ ${\ displaystyle 1}$, ${\ displaystyle 4 \ pi}$ ${\ displaystyle 1}$, ${\ displaystyle 1}$
microscopic Maxwell
equations
${\ displaystyle {\ vec {\ nabla}} {\ vec {E}} = K \ rho}$
${\ displaystyle {\ frac {1} {\ varepsilon _ {0}}}}$ ${\ displaystyle 4 \ pi}$ ${\ displaystyle 4 \ pi c ^ {2}}$ ${\ displaystyle 4 \ pi}$ ${\ displaystyle 1}$
${\ displaystyle {\ vec {\ nabla}} {\ vec {B}} = 0}$ - - - - -
${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {E}} = - K {\ frac {\ partial {\ vec {B}}} {\ partial t}}}$ ${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle c ^ {- 1}}$ ${\ displaystyle c ^ {- 1}}$
${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {B}} = K_ {a} {\ vec {J}} + K_ {b} {\ frac {\ partial {\ vec {E}} } {\ partial t}}}$ ${\ displaystyle \ mu _ {0}}$, ${\ displaystyle \ mu _ {0} \ varepsilon _ {0} = c ^ {- 2}}$ ${\ displaystyle 4 \ pi c ^ {- 2}}$, ${\ displaystyle c ^ {- 2}}$ ${\ displaystyle 4 \ pi}$, ${\ displaystyle c ^ {- 2}}$ ${\ displaystyle 4 \ pi c ^ {- 1}}$, ${\ displaystyle c ^ {- 1}}$ ${\ displaystyle c ^ {- 1}}$, ${\ displaystyle c ^ {- 1}}$

## Electromagnetic units in different systems

Electromagnetic SI units and the corresponding units in three variants of the CGS system
Electromagnetic
quantity
unit Gaussian unit in cgs
SI ESU EMU Gauss
charge Q 1 C 10 −1 c statC 10 −1 ABC 10 −1 c Fr. Fr = statC = g 1/2 cm 3/2 s −1
Amperage I. 1 A 10 −1 c statA 10 −1 abA 10 −1 c statA statA = g 1/2 cm 3/2 s −2
tension U 1 V 10 8 c −1 statV 10 8 abV 10 8 c −1 statV statV = g 1/2 cm 1/2 s −1
electric field strength E. 1 V / m 10 6 c −1 statV / cm 10 6 abV / cm 10 6 c −1 statV / cm statV / cm = g 1/2 cm −1/2 s −1
electric dipole moment p 1 C · m 10 1 c statC · cm 10 1 abC cm 10 19 c D. D = g 1/2 cm 5/2 s −1
magnetic flux density B. 1 T 10 4 c −1 instead of 10 4 G 10 4 G G = g 1/2 cm −1/2 s −1
magnetic field strength H 1 A / m 4π · 10 −3  c statA / cm 4π · 10 −3 Oe 4π · 10 −3 Oe Oe = g 1/2 cm −1/2 s −1
magnetic dipole moment m, μ A · m 2 10 3 c statA cm 2 10 3 abA cm 2 10 3 erg / G G = g 1/2 cm 5/2 s −1
magnetic flooding Θ 1 A 4π · 10 −1 c statA 4π · 10 −1 abA 4π · 10 −1 Gb Gb = g 1/2 cm 1/2 s −1
magnetic river Φ 1 Wb 10 8 c −1 statT cm 2 10 8 G cm 2 10 8 Mx Mx = g 1/2 cm 3/2 s −1
resistance R. 1 Ω 10 9 c −2 s / cm 10 9 abΩ 10 9 c −2 s / cm cm −1 s
specific resistance ρ 1 Ω · m 10 11 c −2 s 10 11 abΩ cm 10 11  c −2 s s
capacity C. 1 F. 10 −9 c 2 cm 10 −9 abF 10 −9 c 2 cm cm
Inductance L. 1 H. 10 9 c −2 cm −1 s 2 10 9 fromH 10 9 c −2 cm −1 s 2 cm −1 s 2
electrical power P 1 V * A = 1 W = 10 7 erg / s 10 7 erg / s 10 7 erg / s erg / s = g cm 2 s −3

The "≙" symbol indicates that this is not a simple conversion of units of measure. The CGS sizes generally have a different dimension than the corresponding size in the SI . That is why it is usually not allowed to simply replace the units in formulas. c is the speed of light .

## literature

• John David Jackson: Classical Electrodynamics. Appendix on Units and Dimensions (also published in German under the title Classical Electrodynamics ).